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The following question was motivated by my research.

Consider a $n\times n$ matrix whose elements are $0$'s or $1$'s such that the determinant is odd. The question is: is it possible to assign signs to matrix elements such that the determinant of the matrix will be equal to $1$? I do not know an answer even to a weaker question: is it possible to replace some of the $1$'s in the matrix with odd integers so that the determinant will be equal to $1$?

Remark: it is known that a natural reduction mod $N$ map $SL_n(\mathbb Z) \to SL_n(\mathbb Z/ N\mathbb Z)$ iz is surjective for any $n,N$.

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# Lifting matrices mod 2 to integers.

The following question was motivated by my research.

Consider a $n\times n$ matrix whose elements are $0$'s or $1$'s such that the determinant is odd. The question is: is it possible to assign signs to matrix elements such that the determinant of the matrix will be equal to $1$? I do not know an answer even to a weaker question: is it possible to replace some of the $1$'s in the matrix with odd integers so that the determinant will be equal to $1$?

Remark: it is known that a natural reduction mod $N$ map $SL_n(\mathbb Z) \to SL_n(\mathbb Z/ N\mathbb Z)$ iz surjective for any $n,N$.